Dynamic models and solutions for evaluating ventilation, perfusion, and mass transfer in the lung--Part I: The models

Modelling mixing within the dead space of the lung improves predictions of functional residual capacity

Routine estimation of functional residual capacity (FRC) in ventilated patients has been a long held goal, with many methods previously proposed, but none have been used in routine clinical practice. This paper proposes three models for determining FRC using the nitrous oxide concentration from the entire expired breath in order to improve the precision of the estimate. Of the three models proposed, a dead space with two mixing compartments provided the best results, reducing the mean limits of agreement with the FRC measured by whole body plethysmography by up to 41%. This moves away from traditional lung models, which do not account for mixing within the dead space. Compared to literature values for FRC, the results are similar to those obtained using helium dilution and better than the LUFU device (Dräger Medical, Lubeck, Germany), with significantly better limits of agreement compared to plethysmography.

A fast, digitally controlled flow proportional gas injection system for studies in lung function

The aim of this paper is to describe a device for flow proportional injection of tracer gas in the lungs of mechanically ventilated patients. This device may then be used for the study of the multiple breath indicator gas washout technique to determine the end-expiratory lung volume. Such a tracer gas injection device may also be used in the study of other techniques that rely on uptake and elimination of tracer gas by the lungs. In this paper, an injector is described which enables injection of indicator gas at a predetermined concentration in a breathing circuit independent of the type of breathing. The presented setup uses a control computer to produce steering signals to a multivalve array in proportion to the input breathing signals. The multivalve array consists of ten circular valves, each with a different diameter, which can be opened or closed individually according to the input signal of the array. By opening of a certain combination of valves an amount of sulphur hexafluoride gas proportional to the inspiratory breathing signal is released. The rate of transmission between the components of the injection system was 80 Hz. The injector has a full flow range between 0-10 L/min. The delay time between the breathing signal and the flow response was 70 ms. The aimed washin gas concentration of 1% SF6 was achieved after 0.5 s. The study describes the results of tests to determine valve-flow ratios, step response and dynamic response of the injector. The flow output response of the injector system was shown to increase in input frequencies above 3 Hz. The valve flow ratios showed the largest relative deviation in the two smallest valves of the 10 valve array, respectively 0.005 L/min (25%) and 0.002 L/min (20%). We conclude that the injector can achieve a stable concentration of indicator gas in a breathing system with an accuracy of 0.005 L/min to execute the multiple breath indicator washout test in human subjects. The results of the study indicate that the injector may be of use in other application fields in respiratory physiology in which breathing circuit injection of indicator gas is required.

A mathematical model of CO2 variation in the ventilated neonate

A mathematical model of the variation of partial pressure of carbon dioxide in the arterial blood of a ventilated neonate is developed. The model comprises alveolar, arterial, pulmonary, venous and tissue compartments, with gas exchange in the lung determined by inspiration and expiration terms. Gas exchange is modelled through diffusion and convective transfer. Carbon dioxide is produced in the tissue by a metabolic term. Shunting is modelled by allowing blood flow to bypass the pulmonary compartment in which diffusion takes place. The model predicts changes in the carbon dioxide partial pressures that occur following abrupt changes in the ventilation settings, and show broad agreement with actual data obtained from novel sensing technology.

A tidal breathing model of the inert gas sinewave technique for inhomogeneous lungs

The tidal breathing model conservation of mass equations for the sinewave technique have been described for a homogeneous alveolar compartment by Gavaghan and Hahn, 1996 [Gavaghan, D.J., Hahn, C.E.W., 1996. A tidal breathing model of the forced inspired gas sinewave technique. Respir. Physiol. 106, 209-221]. We develop these equations first to a multi-discrete alveolar compartment lung model and then to a lung model with a continuous distribution of volume, ventilation and perfusion. The effect on the output parameters of a multi-compartment model is discussed, and the results are compared to those derived from the conventional continuous-ventilation model. Using the barely soluble gas argon as the tracer gas, an empirical index of alveolar inhomogeneity is presented which uses the end-expired and mixed-expired partial pressures on each breath. This index distinguishes between a narrow unimodal distribution of ventilation-volume, a wide unimodal distribution of ventilation-volume and a bimodal distribution of ventilation-volume. By using Monte Carlo simulations, this index is shown to be stable to experimental error of realistic magnitude.

Modelling inert gas exchange in tissue and mixed-venous blood return to the lungs

Inert gas exchange in tissue has been almost exclusively modelled by using an ordinary differential equation. The mathematical model that is used to derive this ordinary differential equation assumes that the partial pressure of an inert gas (which is proportional to the content of that gas) is a function only of time. This mathematical model does not allow for spatial variations in inert gas partial pressure. This model is also dependent only on the ratio of blood flow to tissue volume, and so does not take account of the shape of the body compartment or of the density of the capillaries that supply blood to this tissue. The partial pressure of a given inert gas in mixed-venous blood flowing back to the lungs is calculated from this ordinary differential equation. In this study, we write down the partial differential equations that allow for spatial as well as temporal variations in inert gas partial pressure in tissue. We then solve these partial differential equations and compare them to the solution of the ordinary differential equations described above. It is found that the solution of the ordinary differential equation is very different from the solution of the partial differential equation, and so the ordinary differential equation should not be used if an accurate calculation of inert gas transport to tissue is required. Further, the solution of the PDE is dependent on the shape of the body compartment and on the density of the capillaries that supply blood to this tissue. As a result, techniques that are based on the ordinary differential equation to calculate the mixed-venous blood partial pressure may be in error.

Estimation of pulmonary blood flow from sinusoidal gas exchange during anaesthesia: a theoretical study

We simulated the use of simultaneous sinusoidal changes of inspired O2 and N2O (Williams et al., J Appl Physiol, 1994; 76: 2130-9) at fractional concentrations up to 0.3 and 0.7, respectively, to estimate FRC and pulmonary blood flow (PBF) during anaesthesia, using O2 as an insoluble indicator. Hahn's approximate equations, which neglect the effect of pulmonary uptake and excretion on expiratory flow, estimate dead space and alveolar volume (VA) with systematic errors less than 10%, but yield systematic errors in PBF which are approximately proportional to FIN2O in magnitude. A correction factor (1 - P)-1 for Hahn's equations for PBF (where P is the mean partial pressure of the soluble indicator) reduces the dependence of PBF estimates of FIN2O, and the solution of equations describing the simultaneous mass balance of both indicators yields accurate results for a wide range of mean FIN2O. However, PBF estimates are sensitive to measurement errors and a third gas must be present to ensure that the indicator gases behave independently.

A tidal breathing model of the forced inspired inert gas sinewave technique

We have shown previously that it is possible to assess the cardio-respiratory function using sinusoidally oscillating inert gas forcing signals of nitrous oxide and argon (Hahn et al., 1993). This method uses an extension of a mathematical model of respiratory gas exchange introduced by Zwart et al. (1976), which assumed continuous ventilation. We investigate the effects of this assumption by developing a mathematical model using a single alveolar compartment and incorporating tidal ventilation, which must be solved using numerical methods. We compare simulated results from the tidal model with those from the continuous model, as the governing ventilatory and cardiac parameters are varied. The mathematical model is designed to be the basis of an on-line, non-invasive, cardio-respiratory measurement method, and will only be useful if the associated parameter recovery techniques are both reliable and robust. We demonstrate, in the presence of simulated measurement errors, that: (a) accurate recovery of the ventilatory parameters end-tidal volume, VA, and airways series dead-space, VD, are possible using the tidal breathing model; and (b) that a robust technique for recovery of pulmonary blood flow, QP, can be obtained using the more familiar continuous ventilation model.

A mathematical evaluation of the alveolar amplitude response technique

The underlying mathematical model of the forcing sinewave alveolar amplitude response technique (AART) for measuring lung volume and perfusion is investigated. Making use of numerical techniques, we are able to to evaluate the effects of several assumptions which are implicit in the original technique introduced by Zwart et al., J. Appl. Physiol. 41: 419-429, 1976, and development by several other workers. In particular we are able to show that AART is appropriate for gases of a wider range of solubilities than originally suggested, allowing it to be used with agents, such as nitrous oxide, which are more clinically acceptable. In addition, we are able to show that the effects of recirculation times are likely to be very small using figures for standard man. A least squares parameter recovery technique proves to be very robust to simulated measurement errors and is used to quantify the effects of the modelling assumptions.

On-line determination of pulmonary blood flow using respiratory inert gas analysis

An inert gas analysis method has been developed to perform on-line real time determination of pulmonary blood flow using a nonrebreathing approach. This technique is based on a mathematical model describing mass balance of two inert gases which are breathed using an open gas circuit. The measurements using this method are noninvasive, easy to peform, and do not disturb normal physiological processes. As well, since data are collected on a breath-by-breath basis, it is possible to estimate other respiratory, cardiopulmonary, and metabolic parameters simultaneously in a breath-by-breath manner. Special consideration was given to developing effective data processing algorithms to minimize the influence of measurement noise and respiratory variations. Experimental studies to compare this method with other accepted techniques were conducted to validate the present technique.

An extended soluble gas exchange model for estimating pulmonary perfusion--I: Derivation and implementation

A dynamic model for respiratory exchange of blood soluble gas is described. This model includes a general treatment of tidal breathing, an inhomogeneous lung comprising multiple distensible compartments, and nonlinearities due to multiple-gas effects. The motivation for this new model is the continuing interest in estimating pulmonary perfusion from measurements of respiratory soluble gas exchange. Numerical simulation can be employed to investigate the errors that result from simplifications made in the derivations of simpler models used for this purpose. Examples of such simplifications are the assumptions that ventilation is constant and unidirectional, and that multiple soluble gases can be independently modeled. These results can delimit the boundaries within which perfusion estimates can be considered reliable. An example demonstrating the model and its numerical solution is presented.